Competition Graphs of Strongly Connected and Hamiltonian Digraphs

Abstract

Competition graphs were rst introduced by Joel Cohen in the study of food webs and have since been extensively studied. Graphs which are the competition graph of a strongly connected or Hamiltonian digraph are of particular interest in applications to communication networks. It has been previously established that every graph without isolated vertices (except K 2) which is the competitiongraph of a loopless digraph is also the competitiongraph of a strongly connecteddigraph. We establish an analogous result for one generalization of competition graphs, the p-competition graph. Furthermore, we establish some large classes of graphs, including trees, as the p-competition graph of a loopless Hamiltonian digraph. and show that interval graphs on n 4 vertices are the 2-competition graphs of loopless Hamiltonian digraphs.

A class of acyclic digraphs with interval competition graphs

“…After Kim and Roberts [10] characterized competition graphs of semiorders and interval orders, they introduced acyclic digraphs satisfying C(p) and C * (p) as generalizations of (1,6) h (4,0) v (2 , 5) w (4,4) p (3,3) q (3,1) s (1,2) t(0, 4) x (4,3) y (5,2) z(5,1) semiorders. Then Roberts (pers.…”

“…2 for example.) u (1,6) h (4,0) v (2,5) w (4,4) p (3,3) q (3,1) s (1,2) t(0, 4) x (4,3) y (5,2) z(5,1) …”

“…There have been a number of papers about competition graphs of specific classes of digraphs. For instance, competition graphs of acyclic digraphs have been studied in [1,19], of arbitrary digraphs with or without loops in [1,12], of strongly connected digraphs in [5], of Hamiltonian digraphs in [5,7], of interval digraphs in [11], for various classes of symmetric digraphs in [14][15][16], of semiorders, of acyclic digraphs satisfying property C(p), and of acyclic digraphs satisfying property C * (p) in [10]. By means of extending the results obtained in [10], Roberts (pers.…”

Competition Graphs of Hamiltonian Digraphs

“…The competition graph or conflict graph C(D) has the same vertex set as D, and an edge {u, v}, u = v, if there is a vertex w such that (u, w) and (v, w) are arcs in D. Competition graphs are useful in the study of such diverse systems as food webs and radio networks and have received substantial attention over the past fifteen years. Characterizations of competition graphs for a variety of classes of digraphs have been reported over the years; recently, K. F. Fraughnaugh et al [2] have given characterizations for competition graphs of strongly connected digraphs and hamiltonian digraphs. Here we improve their characterizations of competition graphs of hamiltonian digraphs.…”

“…We use circuit to mean a directed cycle in a digraph. Fraughnaugh [2] contains the following two characterizations. Theorem 1.…”